The strange meaning of complex delta

[lokofer]

The strange meaning of .."complex delta"..

Let's suppose we introduce the function..

$$\delta (a+ix)=f(x)$$ a a real number.. then following the representation..

$$\frac{sin (Nx)}{x}$$ f(x) is oo elsewhere...except for pure complex numbers.

$$f(x)(2\pi) =\int_{-\infty}^{\infty}du e^{iua}exp(ux)$$

f(x) is the "laplace inverse" transform of cos(as)

My question is if we can work or manipulate such function f(x) defined in the post although it makes no or little sense even considering it a "distribution"...

 

[lokofer]

- Oh by the way if you have the usual delta function $$\delta (x)$$:

a) can you define the "composite" function $$F[ \delta (x)]$$ where F is a Real function

b) Is there an "F" so $$F[\delta (x)]= x$$?

c) Can you define the $$D^{a} \delta (x)$$ where a is a Real or complex number ("Fractional calculus" with delta functions )

d) If you had an analytic function F so for every x or z( complex):

$$F(z)= \sum_{n=0}^{\infty} \frac{a(n)}{n!}z^{n}$$

Could we make $$F[\delta (x)]= \sum_{n=0}^{\infty} \frac{a(n)}{n!}[\delta(x)]^{n}$$ ?